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On the
$K(\unicode[STIX]{x1D70B},1)$-problem for restrictions of complex reflection arrangements
Published online by Cambridge University Press: 20 January 2020
Abstract
Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and
$\mathscr{A}(W)$ the set of the mirrors of the complex reflections in
$W$. It is known that the complement
$X(\mathscr{A}(W))$ of the reflection arrangement
$\mathscr{A}(W)$ is a
$K(\unicode[STIX]{x1D70B},1)$ space. For
$Y$ an intersection of hyperplanes in
$\mathscr{A}(W)$, let
$X(\mathscr{A}(W)^{Y})$ be the complement in
$Y$ of the hyperplanes in
$\mathscr{A}(W)$ not containing
$Y$. We hope that
$X(\mathscr{A}(W)^{Y})$ is always a
$K(\unicode[STIX]{x1D70B},1)$. We prove it in case of the monomial groups
$W=G(r,p,\ell )$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this
$K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.
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- Research Article
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- Copyright
- © The Authors 2020